Inexact Newton Method for General Purpose Reservoir Simulation

Inexact Newton Methods are widely used to solve systems of nonlinear equations. The convergence of these methods is controlled by the relative linear tolerance, $\eta_\nu$, that is also called the forcing term. A very small $\eta_\nu$ may lead to oversolving the Newton equation. Practical reservoir simulation uses inexact Newton methods with fixed forcing term, usually in the order of $10^{-3}$ or $10^{-4}$. Alternatively, variable forcing terms for a given inexact Newton step have proved to be quite successful in reducing the degree of oversolving in various practical applications. The cumulative number of linear iterations is usually reduced, but the number of nonlinear iterations is usually increased. We first present a review of existing inexact Newton methods with various forcing term estimates and then we propose improved estimates for $\eta_\nu$. These improved estimates try to avoid as much as possible oversolving when the iterate is far from the solution and try to enforce quadratic convergence in the neighbourhood of the solution. Our estimates reduce the total linear iterations while only resulting in few extra Newton iterations. We show successful applications to fully-coupled three-phase and multi-component multiphase models in isothermal and thermal steam reservoir simulation as well as a real deep offshore west-African field with gas re-injection using the reference CPR-AMG iterative linear solver.